Busemann function on Hyperbolic space - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T17:14:33Z http://mathoverflow.net/feeds/question/109716 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109716/busemann-function-on-hyperbolic-space Busemann function on Hyperbolic space jiangsaiyin 2012-10-15T13:28:26Z 2012-10-15T15:35:21Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/109697/laplacian-of-busemann-function-on-hyperbolic-space" rel="nofollow">laplacian of Busemann function on hyperbolic space</a> </p> </blockquote> <p>What's the laplacian of the Buseman function on Hyperbolic space H^n?=n-1?When restricted to geodesics,is it linear?And the level sets are totally geodesic?</p> http://mathoverflow.net/questions/109716/busemann-function-on-hyperbolic-space/109728#109728 Answer by unknown (google) for Busemann function on Hyperbolic space unknown (google) 2012-10-15T15:30:08Z 2012-10-15T15:30:08Z <p>Heintze-ImHof "Geometry of Horospheres" contains a proof that the Busemann functions are C^2. </p> <p>When F is a Busemann function associated to a point z in the ideal boundary, then Z:=-grad(F) is the vectorfield showing towards z and the derivative of Z in direction v is Y'(0), where Y is the Jacobi field with Y(0)=v. </p> <p>The level sets of Busemann functions are horospheres, in some sense the opposite from totally geodesic submanifolds.</p> <p><a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.jdg/1214434219" rel="nofollow">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.jdg/1214434219</a></p>